UDC 513.573.3

MATHEMATICS

E. V. GAIDUKOV

ASYMPTOTIC GEODESICS ON A RIEMANNIAN MANIFOLD NON-HOMEOMORPHIC TO THE SPHERE

(Presented by Academician A. N. Kolmogorov on 25 XI 1965)

§ 1. Formulation of the result. 1. As the simplest application of Theorem 1 proved below, let us consider the plane double pendulum. Its position is determined by two angular coordinates \(\varphi\) and \(\psi\); the configuration space of the double pendulum is the two-dimensional torus. Applying Theorem 1 to the case of the two-dimensional torus, we obtain the following result:

Whatever the initial values \(\varphi_0\) and \(\psi_0\) of the coordinates \(\varphi\) and \(\psi\) may be, one can choose such initial velocities \(\dot\varphi_0\) and \(\dot\psi_0\) that the motion of the pendulum determined by the initial conditions \(((\varphi_0,\psi_0,\dot\varphi_0,\dot\psi_0)\) will asymptotically approach such a periodic motion in which one of the links of the pendulum makes \(k\) revolutions and the other \(l\) revolutions during one and the same fixed interval of time (\(k\) and \(l\) are arbitrarily prescribed integers).

An analogous assertion is valid for any dynamical system with two degrees of freedom.

1. Everywhere in what follows, the letter \(M\) denotes a smooth oriented two-dimensional Riemannian manifold, complete in its metric \((^1)\). All curves on \(M\) are assumed to be piecewise smooth. Two curves are said to be freely homotopic on the manifold \(M\) if they can be continuously deformed into one another without leaving the manifold \(M\) during the deformation (for the precise definition see \((^2)\)).

Theorem 1. Let \(x\) be an arbitrary point of a compact manifold \(M\) not homeomorphic to the two-dimensional sphere; let \(\gamma\) be an arbitrary class of freely homotopic closed paths of the manifold \(M\). Then there exists a half-geodesic \(L\) issuing from the point \(x\), asymptotic to some closed geodesic \(\Pi\) of the class \(\gamma\).

The proof of this theorem is based on simple geometric arguments set forth in the following paragraphs.

§ 2. Minimal curves. In this paragraph we shall formulate several well-known propositions concerning properties of minimal curves which will be used below.

1. A curve \(\Gamma\) on the manifold \(M\) is called minimal between two points \(x,y\in\Gamma\) if the segment \(\Gamma(x,y)\) of the curve \(\Gamma\) enclosed between the points \(x\) and \(y\) has length not exceeding the length of any curve on \(M\) joining the points \(x\) and \(y\). A minimal curve is a curve that is minimal between any two of its points.

2. Let \(\Lambda_1\) and \(\Lambda_2\) be two minimal curves with endpoints \(x_1,y_1\) and \(x_2,y_2\), respectively. Then only the following cases are possible:
   a) the curves \(\Lambda_1\) and \(\Lambda_2\) intersect in no more than one point;
   b) the curves \(\Lambda_1\) and \(\Lambda_2\) are continuations of one another; in this case their intersection is nonempty and is a minimal curve (this intersection may consist of one point);
   c) the curves \(\Lambda_1\) and \(\Lambda_2\) coincide;
   d) the curves \(\Lambda_1\) and \(\Lambda_2\) intersect only at their common endpoints.

1. From item 2 there follows an assertion important for what follows:

Lemma 1. Two minimal curves with a common initial point at a point \(x \in M\) intersect in no more than one point distinct from \(x\), except for the case in which one of these curves is entirely contained in the other.

§ 3. Minimal geodesics on a surface of genus \(>0\). Beginning with this paragraph, the letter \(M\) denotes a compact surface not homeomorphic to the sphere.

1. The universal covering manifold of \(M\) is the two-dimensional plane, endowed with the induced metric; the projection \(p:\widetilde M \to M\) is locally isometric. The following assertions are valid:

a) If \(\Lambda\) is a minimal arc on \(M\), then the covering path \(\widetilde\Lambda\) is minimal on \(\widetilde M\).

Fig. 1

b) Let \(\Gamma\) be a minimal geodesic of the class \(\gamma\) of freely homotopic paths of the manifold \(M\). Then the curve \(\Gamma\) is a minimal loop at each of its points; the preimage of each such loop is a minimal arc on \(\widetilde M\) with endpoints \(\widetilde y_1\) and \(\widetilde y_2\): \(p(\widetilde y_1)=p(\widetilde y_2)\).

1. From 1a) it follows directly:

The curve \(\widetilde\Gamma\), lying over the minimal closed geodesic \(\Gamma \in \gamma\), is a geodesic on \(\widetilde M\).

1. Lemma 2. The geodesic \(\widetilde\Gamma\) of item 2 is minimal between any two of its points.

We preface the proof of the lemma with the following simple observation. The fundamental group \(\pi_1(M)\) of the manifold \(M\) is naturally isomorphic to the group of all motions of the universal covering \(\widetilde M\) lying over the identity motion of the manifold \(M\). By virtue of this isomorphism we identify motions of the covering with elements of the group \(\pi^1\) and denote them by the same letters.

1. Proof of Lemma 2. Let \(\Gamma \in \gamma\) be a minimal geodesic and let \(\widetilde\Gamma\) lie over \(\Gamma\). Let \(\widetilde y\) be an arbitrary point on the geodesic \(\widetilde\Gamma\). Put
   \[ \widetilde y_k=\gamma^k(\widetilde y),\quad k=0,\pm1,\pm2,\ldots \]
   (Fig. 1).

Each of the segments \(\widetilde\Gamma(\widetilde y_k,\widetilde y_l)\) of the geodesic \(\widetilde\Gamma\), enclosed between the points \(\widetilde y_k\) and \(\widetilde y_l\), is minimal (1b). We shall prove that the segment \(\widetilde\Gamma(\widetilde y,\widetilde y_2)\) is minimal. If this is not so, then there exists a minimal arc \(S\) joining the points \(\widetilde y\) and \(\widetilde y_2\), having length strictly less than the length of the arc \(\widetilde\Gamma(\widetilde y,\widetilde y_2)\). Put \(S_1=\gamma(S)\), \(S_{-1}=\gamma^{-1}(S)\) (Fig. 1). The three minimal curves \(S\), \(S_1\), and \(S_{-1}\) obviously intersect at certain points \(\widetilde u\) and \(\widetilde v\) (see Fig. 1).

Denote the length of the arc \(\widetilde\Gamma(\widetilde y,\widetilde y_1)\) by \(|\gamma|\), and the distance function by \(\rho\). We have
\[ \rho(\widetilde y,\widetilde u)+\rho(\widetilde v,\widetilde y_2)\ge |\gamma|. \]
At the same time:

a) \(\rho(\widetilde u,\widetilde v)\ge |\gamma|\), for \(\Gamma\) is minimal in its class of free homotopy;

b) the length \((S)=\rho(\widetilde y,\widetilde u)+\rho(\widetilde u,\widetilde v)+\rho(\widetilde v,\widetilde y_2)\ge 2|\gamma|\), i.e., the length of the arc \(S\) is not less than \(2|\gamma|\), contrary to the supposition. The proof of the lemma is now easily completed by induction.

§ 4. Construction of an asymptotic ray on the covering.

1. Let \(\widetilde y\) and \(\widetilde y'\) be two distinct points lying over some point \(y \in \Gamma\). Let, furthermore, \(\widetilde\Gamma\) and \(\widetilde\Gamma'\) be minimal geodesics passing through the points \(\widetilde y\) and \(\widetilde y'\), respectively, and covering the minimal closed geodesic \(\Gamma\) of the class \(\gamma\).

Let \(T\) be some minimal arc joining the points \(\tilde y\) and \(\tilde y'\). Put \(T_k=\gamma^k(T)\), \(k=0,\pm1,\pm2,\ldots\) (Fig. 2).

1. Choose a point \(\tilde x\) over the given point \(x\in M\), lying in the strip \(Q\) between the geodesics \(\widetilde\Gamma\) and \(\widetilde\Gamma'\). Let \(\widetilde\Lambda_k\), \(k=0,1,2,\ldots\), be some minimal arcs joining the point \(\tilde x\) to the points \(\tilde y_k\). Denote by \(\xi_k\) the linear element of the arc \(\widetilde\Lambda_k\) at the point \(\tilde x\). An immediate consequence of Lemma 1 is the following assertion:

The vector \(\xi_k\) rotates monotonically; consequently, there exists a limiting direction:
\[ \xi_\infty=\lim_{k\to+\infty}\xi_k . \]

1. Let \(\widetilde L\) be the half-geodesic issuing from the point \(\tilde x\) in the direction \(\xi_\infty\). From the properties of minimal curves (§ 2) it follows that:

a) The half-geodesic \(\widetilde L\) is minimal between any two of its points, i.e. it is a geodesic ray ([3]).

b) The geodesic ray \(\widetilde L\) does not intersect the curve \(\Gamma\).

1. Let \(\alpha_k\) be the coordinate of the point of intersection of the curve \(\widetilde L\) with the arc \(T_k\), measured from the point \(\tilde y_k\) along the arc \(T_k\) (we take the coordinate of the point \(\tilde y_k\) to be zero, and the length of the arc \(T_k\) to be one).

Lemma 3. The sequence \(\alpha_k\) is monotone.

It is enough to establish the monotonicity of the corresponding sequence for each of the curves \(\widetilde\Lambda_i\). This is easily done with the aid of arguments analogous to those given in § 3, item 4.

Fig. 2

1. Consider the sequence of iterations \(\widetilde L^{-n}=\gamma^{-n}(\widetilde L)\) of the geodesic ray \(\widetilde L\). From the preceding lemma the following follows.

Main assertion. The sequence of geodesic rays \(\widetilde L^{-n}\) converges to a certain geodesic \(\widetilde\Pi\) on the manifold \(\widetilde M\). The geodesic \(\widetilde\Pi\) is minimal between any two of its points and is invariant with respect to the motion \(\gamma\). Moreover, the geodesic ray \(\widetilde L\) is asymptotic to the geodesic \(\widetilde\Pi\).

1. Projecting the geodesic \(\widetilde\Pi\) and the ray \(\widetilde L\) onto the manifold \(M\), we obtain a closed geodesic \(\Pi\) of the class \(\gamma\) and a half-geodesic \(L\), asymptotic to \(\Pi\). Theorem 1 is proved.

2. Remark. Generally speaking, the limiting geodesic \(\Pi\) need not coincide with the initial closed geodesic \(\Gamma\). Nor is it necessarily minimal in the class \(\gamma\). However, the following assertion is true:

The geodesic \(\Pi\) and the asymptote \(L\) do not contain conjugate points.

Moscow State University
named after M. V. Lomonosov

Received
17.XI.1965

REFERENCES

1. H. Hopf, W. Rinow, Comm. math. Helv., 3, 209 (1931).
2. G. Seifert, W. Threlfall, Topology, Moscow–Leningrad, 1938.
3. S. E. Cohn-Vossen, Shortest paths and problems of curvature, in the book: S. E. Cohn-Vossen, Some problems of differential geometry in the large, Moscow, 1959.